Question

A circle has a center at 4 – 5i and a point on the circle at 19 – 13i. Which of the following points is also on the circle?

Answer 1

Hers is a solution : 

Answer 2

Answer:

The correct option is  3.

Step-by-step explanation:

The center of the circle is (4-5i). A point on the circle at (19 – 13i).

Distance formula between two points $z_1=x_1+iy_1\text{ and } z_2=x_2+iy_2$ is

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

The radius of the circle is

$d=\sqrt{(19-4)^2+(-13-(-5))^2}=17$

The radius of the circle is 17 units.

The distance between –11 –3i and (4-5i).

$d=\sqrt{(-11-4)^2+(-4-(-5))^2}=\sqrt{226}$

The distance between –4.5+3.5i and (4-5i).

$d=\sqrt{(-4.5-4)^2+(3.5-(-5))^2}=12.02$

The distance between -12+10i and (4-5i).

$d=\sqrt{(12-4)^2+(10-(-5))^2}=17$

The distance between -12+10i and (4-5i) is equal to the radius. Therefore point 12+10i lies on the circle.

The distance between 21+12i and (4-5i).

$d=\sqrt{(21-4)^2+(12-(-5))^2}=21.93$

Therefore option 3 is correct.